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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Substituting <span class="process-math">\(u(x, t)=XT\)</span> in the boundary condition at <span class="process-math">\(x = 0\text{,}\)</span> we obtain</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(0,t)=X(0)T(t)=0.
\end{equation*}
</div>
<p class="continuation">If above equation is satisfied by choosing <span class="process-math">\(T(t)\)</span> to be zero for all <span class="process-math">\(t\text{,}\)</span> then <span class="process-math">\(u(x,t)\)</span> is zero for all <span class="process-math">\(x\)</span> and <span class="process-math">\(t\text{,}\)</span> and we have already rejected this possibility. Therefore it must be satisfied by requiring that</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
X(0)=0.
\end{equation*}
</div>
<p class="continuation">Similarly, the boundary condition at <span class="process-math">\(x = L\)</span> requires that</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
X(L)=0.
\end{equation*}
</div>
<p class="continuation">We have the following <em class="emphasis">eigenvalue problem</em> for <span class="process-math">\(X(x)\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
X''+\lambda X=0,\quad X(0)=X(L)=0,
\end{equation*}
</div>
<p class="continuation">which is an example we had earlier. The nontrivial solutions should be eigenfunctions</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation}
{\lambda_n}={\left(\frac{n\pi}{L}\right)^2},\quad {X_n(x)}={\sin\frac{n\pi x}{L}},\quad n={1},2,3,\cdots\smallskip\tag{7.7.1}
\end{equation}
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